Simplify $A=\frac{1}{\left(a^{-\frac12}-a^\frac23\right)^3}\left(\frac{1}{a}-2a^\frac16+a^\frac43\right)^\frac32$

Question

Simplify $$A=\dfrac{1}{\left(a^{-\frac12}-a^\frac23\right)^3}\left(\dfrac{1}{a}-2a^\frac16+a^\frac43\right)^\frac32, a>0,a\ne1.$$ So let's the first factor be $B$. We have $$B=\dfrac{1}{\left(\dfrac{1}{a^\frac12}-a^\frac23\right)^3}=\dfrac{1}{\left(\dfrac{1-a^\frac76}{a^\frac12}\right)^3}=\dfrac{a^\frac32}{\left(1-a^\frac76\right)^3}$$ Let's the second factor be $C$ (so $A=BC$). I don't see what useful we can do with $C$. Can you give me a hint?

Answer 1

Note that $$a^{-1}-2a^{1/6}+a^{4/3}=(a^{-1/2}-a^{2/3})^2$$ Call the binomial inside the bracket $b$, then $$A=\frac{b^{2×3/2}}{b^3}=\frac{b^3}{b^3}=1$$